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Q1 Are there options that doesn't have a closed form pricing formula like BS? Is this the situation when we have to use Finite Difference Method?

Can someone give an example? (I hope this option really exist so that I can look for its historical data from Yahoo Finance)

Q2 After pricing the option, what should we do to develop trading strategies?

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The question about the availability of closed-form solutions can generally not be answered for types of options alone but only for the combination of a payoff function and the underlying asset dynamics.

Consider for example European plain vanilla options. These have a (quasi) closed-form solution when the underlying follows an exponential Levy processes including geometric Brownian motion (GBM), in many stochastic volatility models but not under local volatility dynamics. However, you will usually find that options that cannot be priced in closed-form under GBM cannot be priced in closed-form under other (sensible) underlying asset dynamics either.

Furthermore, many more exotic options that can be priced in closed-form under GBM should not be priced using this model. The reason is that the value of the payoff is heavily driven by risk-factors whose dynamics are not modelled. Examples are the forward skew sensitivity of American binary options, the stochastic interest rate exposure of equity linked bonds with an uncertain maturity, ... .

Once the option cannot be priced in closed-form under the respective model dynamics, you need a numerical method such as finite differences, Monte Carlo simulation, trees, ... .

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Most of the options in finance don't have closed form solutions. That includes American and many exotic options.

When you don't have a formula, you may use numeric methods such as FDM. You may also approximate the option with an analytic solution.

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